monte carlo simulations of polyion−macroion complexes. 1. equal absolute polyion and macroion...

Monte Carlo Simulations of Polyion-Macroion Complexes. 1. EqualAbsolute Polyion and Macroion Charges

Anna Akinchina* and Per Linse

Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University,Box 124, SE-221 00, Lund, Sweden

Received November 26, 2001; Revised Manuscript Received March 25, 2002

ABSTRACT: Intermolecular structures of complexes formed between a charged polymer and a sphericaland oppositely charged macroion have been studied by employing the primitive model solved by MonteCarlo simulations. The strong-complex case involving a polyion and a macroion with equal absolute chargesand without small ions was considered. The influence of the polyion flexibility with a bare persistencelength ranging from 7 to 1000 Å for four different systems characterized by different polyion linear chargedensities and macroion sizes has been examined. Radial distributions, polyion bead complexationprobabilities, loop, tail, and train characteristics, and energetic analysis have been performed. Thestrongest and most compact complex, involving a collapsed polyion wrapping the macroion, was formedfor a semiflexible chain. As the stiffness was increased, this state was transformed into a range of differentstructures comprising “tennis ball seam”-like, solenoid, multiloop (“rosette”), and single-loop arrangementsas well as structures involving only a single polyion-macroion contact region.

1. IntroductionComplex formation between charged polymers (poly-

ions) and oppositely charged colloids (macroions) is ofgreat interest both in practical applications and intheoretical investigations of modern colloidal science.In nature, polymers such as DNA, polyamines, manypolysaccharides, etc., are charged, and many colloidalparticles such as proteins, aggregates of them, lipo-somes, etc., are charged as well. The number of differentmanmade polyions is extensive, and in technologicalapplications charged surfactant micelles, silica particles,latex particles, just to mention a few charged colloids,are widely used as well. The experimental studies inthis area have a long history; see, e.g., the reviews byGoddard,1 Kwak,2 Lindman and Thalberg,3 Hanssonand Lindman,4 and Doublier et al.5 More recentstudies6-13 have also provided valuable insights.

During the past decade, the adsorption of polyions atoppositely charged surfaces has been subjected to anintense theoretical interest, and different theoreticalapproaches including simulation methods have beenemployed. Most of these studies involve a polyionadsorbing onto (i) planar surfaces representing mono-layer and bilayer surfaces and surfaces of large par-ticles, (ii) cylindrical surfaces representing e.g. histones,and (iii) spherical surfaces representing micelles, vesicles,globular proteins, dendrimeres, etc. Owing to the largenumber of reduced variables in these systems, much ofthe present theoretical developments do not yet overlap,even in the reduced parameter space. Hence, theorieshave not yet to any larger extent been confronted byessentially exact solutions of models provided by simu-lations.

The more recent theoretical investigations of thecomplexation between polyions and oppositely chargedmacroions are now extended,14-34 and some of theearlier studies have also recently been summarized.35

Most of these studies consider the complexation betweenone polyion and one macroion, but also the cases of

several macroions27,31-34 and several polyions18 havebeen considered.

In more detail, Monte Carlo simulations have previ-ously been employed to obtain structural and thermo-dynamical aspects of the polyion-macroion complex-ation. In a series of papers,15-17 Wallin and Linseexamined such complexations applied to surfactantmicelles at different conditions with the focus on ther-modynamic aspects. Further simulations have beenperformed by Kong and Muthukumar19 and Nguyen andShklovskii27 to test previous theoretical predictions.Chodanowski and Stoll have simulated the polyion-macroion complexation and examined how the complexstructure depends on the macroion size28 and polyionchain length29 at different screening lengths. Moreover,Jonsson and Linse have simulated the complexationbetween one polyion and varying amount of macroions,including the limit where the complex becomes satu-rated of macroions at different macroion and polyionextension and charge,31 polyion stiffness,32 and saltcontent.34 In addition, the complexation between apolyion and a lysozyme has been simulated by Carlssonet al.30

The aim of the present study is to obtain furtherunderstanding on the complexation between a polyionand an oppositely charged macroion by performingsystematic Monte Carlo simulations. We have employeda minimal system containing one polyion and onemacroion with equal absolute charges and no small ions.Our focus is how the bare stiffness of the polyion affectsthe complex employing four different systems character-ized by different polyion linear charge densities andsizes of macroion keeping the charges of the twocomponents fixed. Despite the restricted selection ofsystem parameters, a rich structural behavior is ob-tained. For flexible chains we obtain a collapsed polyionlayer, as been considered in, e.g., refs 20, 22, 26, and31, whereas with stiffer chains “tennis ball seam”-likestructure as previously observed by Jeppesen et al.,36

multiloop or “rosette” structures as theoretically pre-dicted by Schiessel et al.,25 or solenoid arrangement as* Corresponding author. E-mail: [email protected].

5183Macromolecules 2002, 35, 5183-5193

10.1021/ma012052u CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 05/21/2002

assumed by Schiessel et al.33 and by Nguyen andShklovskii26 are observed. Our work is also related tothe theoretical predictions by Netz and Joanny,23 wherethey investigated the “touching” and “wrapping” transi-tions, and it also complements the recent simulationstudies by Chodanowski and Stoll.28,29

The paper is organized as follows: Section II describesthe model system and some simulation aspects. Ourresults are presented in section III and discussed insection IV. In section V, we conclude our findings, andthe paper ends with an appendix, where we compareand discuss different methods to calculate the persis-tence length of a polymer.

2. Method

2.1. Model. The complex formed by one polyion andone macroion is described in the framework of theprimitive model and using a cell model. The formerimplies that (i) the charged species (charged polyionsegments and macroions) are represented as chargedhard spheres, each specified by a charge and a hard-sphere radius, and (ii) the solvent is treated as acontinuum with a constant relative permittivity. Thelatter involves a confinement of the species inside aspherical cell, and in the present study the macroionwas also fixed at the center of the cell.

As alluded to in the Introduction, the present systemis described by a large number of parameters. To clarifythe situation and to make comparison with other studiesmore transparent, we will provide a relevant set ofvariables and motivate our choice of systems investi-gated.

A system comprised of one polyion and one macroionwith their counterions and additional salt of the sametype as the counterions described within the primitivemodel requires 15 physical relevant variables. Table 1

displays one such set of variables, their symbols, nu-merical values used, and some other variables derivedfrom these. In the present study, we have focused onsystems (i) in which the polyion and the macroion carrysame absolute charges, (ii) free of small ions, and (iii)where the cell radius Rcell is sufficiently large to makethe volume in practice irrelevant, thus leaving eightvariables left. Moreover, the focus on (iv) aqueoussolution at ambient temperature fixes T and εr and (v)alkyl-based polyions restricts the bead-bead separationand bead size to a narrow window. Finally, only strongcomplexes formed are of interest, and the macroioncharge Zm ) (-Zp )) -40 has been assigned throughout.

The three remaining independent variables are (i) themacroion radius Rm, (ii) the bead charge Zb, and (iii)the chain flexibility here expressed by the bare persis-tence length lP. Table 2 specifies the four differentsystems considered, which are obtained by combiningthe bead charges 0.5 and 1 with the macroion radii 10and 20 Å. The properties of these four systems havebeen investigated at seven different chain flexibilities,ranging from fully flexible chains, lP ) 7 Å, to nearlyrigid rods, lP ) 1000 Å (see also Table 1). As presentedbelow, the variation of these three parameters is suf-ficient to monitor a rich behavior of the structure of thepolyion-macroion complex.

The polyion is represented by a chain of charged beadsconnected by harmonic bonds. The total potential energyfor a system can be expressed as

where the terms describing the hard-core repulsion andthe Coulomb interaction are given by

with Zi and Ri denoting the charge and the radius ofparticle i (the macroion or a polyion bead), e theelementary charge, ε0 the permittivity of vacuum, εr therelative permittivity of water, and rij the distancebetween the centers of particles i and j.

Table 1. Variables of the Systems Investigated

variable symbol value

Generalvolume V largea

temperature T 298 Krelative permittivity εr 78.5Bjerrum lengthb LB ≡ e2/(4πε0εrkT) 7.14 Å

Polyionbead charge Zb (>0) 0.5 and 1c

bead radius Rb 2.0 Åbead-bead separation b ≈5.5 Åno. of beads Nb dbare persistence length lP 7, 30, 60, 120, 250,

500, and 1000 Ålinear charge densityb λp ≡ eZb/bpolyion contour lengthb L ≡ b(Nb - 1)

Macroioncharge Zm (<0) -40radius Rm 10 and 20 Åc

surface charge densityb σm ≡ eZm/(4πRm2)

Small Ionssalt number density FS 0ion charges Z+, Z-ion radii R+, R-

a A spherical cell with radius Rcell ) 1241 Å was used. Theprecise value of the volume is in practice irrelevant, owing to thestrong complexation free energy. For example, the polyion mixingentropy becomes relevant first when the system volume V ≈Vcomplex exp(Acomplex/kT) ≈ 1015 m3, estimated by using a complex-ation volume Vcomplex ) 102 Å3 and a complexation free energyAcomplex/kT ) 100. b Dependent variable. c See Table 2. d Fixed bythe requirement Zp ≡ NbZb ) -Zm.

Table 2. Specific Variables and Some Results of theSystems Investigated

system I II III IV

Independent VariablesZb 0.5 0.5 1 1Rm (Å) 20 10 20 10

Important Dependent VariablesNb 80 80 40 40Γmb ≡ |ZmZb|LB/(Rm + Rb) 6.50 11.9 13.0 23.8contour length L ) b(Nb - 1) (Å)a 430 430 210 210L/[2π(Rb + Rm)] 3.1 5.8 1.6 2.8bead coverage Nb(πRb

2)/(4πRm2) 0.2 0.8 0.1 0.4

bond coverageNb(π(b/2)2)/(4πRm

2)b0.4 1.5 0.2 0.7

Bead-Bead Separation⟨Rbb

2 ⟩1/2 (Å) (lP ) 7 Å) 5.68 5.55 5.91 5.64⟨Rbb

2 ⟩1/2 (Å) (lP ) 60 Å) 5.54 5.50 5.73 5.29⟨Rbb

2 ⟩1/2 (Å) (lP ) 1000 Å) 5.53 5.47 5.39 5.23

a b ) 5.5 Å. b b ) ⟨Rbb2 ⟩1/2 with lP ) 1000 Å.

U ) Uhc + Uel + Ubond + Uangle (1)

Uhc + Uel ) ∑i<j

Uij (2)

Uij ) {∞ rij < Ri + Rj

ZiZje2

4πε0εrrij

rij g Ri + Rj(3)

5184 Akinchina and Linse Macromolecules, Vol. 35, No. 13, 2002

The polymer chain connectivity and flexibility arecontrolled by the bond and angular harmonic potentialenergies according to

and

respectively, where Nb is the number of beads in thepolyion, kbond the bond force constant, ri,i+1 the distancebetween two connected beads, r0 the zero-force bonddistance, kangle the angle force constant, Ri the anglebetween three consecutive beads (bond angle), and R0the zero-force bond angle. Here, we have used r0 ) 5.0Å and kbond ) 0.4 N/m, which in combination with thereaming interactions leads to a root-mean-square (rms)bead-bead separation ⟨Rbb

2 ⟩1/2 ≈ 5.5 Å. Moreover, R0 )180° was selected, and kangle ) 0, 3.6, 7.7, 15.8, 33.5,67.4, and 135 J/(mol deg2) were used, which give thebare persistence lengths (the persistence length calcu-lated for an uncharged chain) displayed in Table 1. Thepersistence length lP was evaluated according to

and a more detailed description of its evaluation ispresented in the Appendix. The chains of differentstiffness will be labeled according to their bare persis-tence lengths.

The strong electrostatic interaction between the poly-ion and the macroion gives rise to a firm complex at allconditions. A central quantity is the reduced electro-static interaction of a polyion bead at hard-spherecontact with the macroion, Γmb ≡ |ZmZb|LB/(Rm + Rb),where LB is the Bjerrum length (see Table 1). Theweakest interaction appears in system I (characterizedby Zb ) 0.5, Nb ) 80, and Rm ) 20 Å) where Γmb ) 6.50.In systems II (Zb ) 0.5, Nb ) 80, and Rm ) 10 Å) andIII (Zb ) 1, Nb ) 40, and Rm ) 20 Å), we have Γmb )11.9 and 13.0, respectively, and the strongest interactionappears in system IV (Zb ) 1, Nb ) 40, and Rm ) 10 Å)where Γmb ) 23.8. These and geometrical data as thepolyion contour length, the polyion length divided bythe circumference of the macroion, and the bead andthe bond coverage are collected in Table 2.

2.2. Simulation Details. All Monte Carlo (MC)simulations were performed in the canonical ensemble,employing the standard Metropolis algorithm.37 Sincethe position of the macroion was fixed, only the polyionbeads were subjected to trial moves. Four different typesof MC trial moves were employed: (i) single bead move,(ii) pivot rotation where the chain is randomly dividedinto two parts and the shortest of them rotated aroundthe point of division, (iii) translation of the entire chain,and (iv) slithering move where one of the end beads ismoved to the opposite end of the chain with biasedradial and angular positioning. The single particle movewas attempted 50 times more often than the other threetypes of moves. The displacement parameters of thedifferent types of trial moves were selected for each casewithin the following ranges: ∆bead ) 0.7-2.5 Å, ∆R )

15°-360°, and ∆chain ) 0.3-0.6 Å. The average ac-ceptance probabilities of the four types of moves were≈0.4, ≈0.4, ≈0.5, and ≈0.6.

Most often the polymer beads were initially placedrandomly in the simulation cell and separated from themacroion fixed at the center, leading to separatedmacromolecules. The simulation of each complex in-volved typically an equilibration of 5 × 105 MC passes(trial moves per particle) followed by a production runof 1 × 106 MC passes. In cases with large barepersistence lengths, it was expected that the equilibriumprocess could be trapped in metastable conformationscharacterized by the number of polyion loop (cf. Figure1). Therefore, systems I and II with lP g 250 Å andsystem IV with lP ) 1000 Å were simulated withdifferent initial configurations. In addition to separatedmacromolecules, also collapsed complexes achieved fromsimulations with shorter persistence lengths were em-ployed as initial configurations. In those cases wheredifferent initial configurations produced states charac-terized by different number of loops (system I with lP )1000 Å and system II with lP ) 500 and 1000 Å), resultsof the state with the lowest potential energy of eachsystem are reported, assuming the entropy differencesamong the states being less important.

Reported errors are one standard deviation, and theyare evaluated by diving the total simulation intotypical 10 subbatches. All the simulations were carriedout with the use of the integrated Monte Carlo/molec-ular dynamics/Brownian dynamics simulation packageMOLSIM.38

3. Results

3.1. Overview. An overview of the structures of thecomplexes obtained is given in Figure 1. The finalconfigurations of the simulations are displayed forsystems I-IV (from left to right) with five of the sevenflexibilities considered (increasing lP from top to bottom).

These configurations illustrate that a polyion-mac-roion complex appears at all conditions, which also isconfirmed by the statistical analysis presented below.Another general observation is that with a fully flexiblechain (top row) all polyion beads are near the macroionin a disordered manner. Furthermore, as the chainstiffness is increased progressively, the chain becomeslocally less folded, and configurations such as a “rosette”(system II, lP ) 250 Å), a “tennis ball seam” (systemIII, lP ) 60 Å), and a solenoid (system IV, lP ) 60-500Å) occur. With the largest stiffness, lP ) 1000 Å (bottomrow), only a limited number of polyion beads appearnear the macroion, and depending on the system asingle loop (systems I, II, and IV) or a U-turn (systemIII) occurs. Moreover, in systems I and II with lP ) 1000Å, metastable conformations (see section 2.2) with aslightly to moderately bent polyion making only a singlecontact with the macroion were also obtained. Here, themacroion was in contact with only a few central beadsforming a bend of the chain.

Hence, the four systems employed, being composedof one polyion and one macroion with same absolutecharges, display a rich sequence of structures as thebare persistence length is altered, and moreover thesequences of structures differ among the systems.

3.2. Complex Structure. 3.2.1. Radial Bead Distri-bution. The number of beads within the distance r fromthe center of the macroion will now be considered. Theso-called running coordination number, rcn(r), has been

Ubond ) ∑i)1

Nb-1kbond

2(ri,i+1 - r0)

2 (4)

Uangle ) ∑i)2

Nb-1kangle

2(Ri - R0)

2 (5)

lP )⟨Rbb

2 ⟩1/2

1 + ⟨cos Ri⟩(6)

Macromolecules, Vol. 35, No. 13, 2002 Polyion-Macroion Complexes 5185

calculated, and the function will furthermore be nor-malized by the number of polyion beads Nb. Hence, thenormalized running coordination number, rcn(r)/Nb,expresses the fraction of beads located within thedistance r from the center of the macroion, and bydefinition rcn(r)/Nb is zero for r < Rm + Rb andapproaches unity at large r.

Figure 2 displays the normalized rcn(r) for each ofthe four systems with chains of different bare persis-tence lengths. Generally, all functions display a sharpincrease at the macroion-bead contact separation r )Rm + Rb and thereafter a leveling off, although thesteepness of the initial rise and the distance at whichthe limiting value is approached differ among thesystems.

In more detail, the initial increase of the reduced rcnis less steep in systems I and II (Zb ) 0.5) as comparedto those in systems III and IV (Zb ) 1). With the

completely flexible chains (lP ) 7 Å), nearly all beadsare within r ) Rm + Rb + 10 Å in the two formersystems, whereas in the two latter systems a 5 Å thicklayer is sufficient. This difference in the thickness ofthe polyion layer at the macroion surface is also visiblein Figure 1.

Moreover, as the stiffness is slightly increased, thesteepness of the reduced rcn increases in systems I, III,and IV. Apparently, the accumulation of the oppositelycharged polyion beads is facilitated by some rigidity ofthe chain, an observation that we will return to. Incontrast, in system II the steepness of the increase ofthe reduced rcn becomes smaller, and the approach tothe limiting value appears at larger distances as thechain stiffness is increasedsfeatures that are con-sistent with the appearance of beads at larger separa-tion from the macroion with increasing lP displayed inFigure 1.

Figure 1. Final configurations from the simulations of systems I-IV (from left to right) with bare persistence lengths lP ) 7, 60,250, 500, and 1000 Å (from top to bottom). The same length scale is used throughout except for systems I with lP ) 1000 Å andsystem II with lP ) 500 and 1000 Å.


3.2.2. Bead Complexation Probability. The configura-tions of the polyion will now be examined more thor-oughly. The division of the beads into two groups,collapsed and noncollapsed, is frequently used in theory,and this division is employed in our analysis as well.Here, we adopt a geometrical definition and consider abead as being collapsed if it is located within 5 Å fromthe closest approach of the macroion surface, i.e., when

the separation between the centers of a bead and of themacroion does not exceed the distance r ) Rc ≡ Rm +Rb + 5 Å. As obvious from Figure 2, there is someambiguity of selecting Rc; however, the qualitativeconclusions are not affected by the precise choice.

The complexation probability function, Pc(ib), provid-ing the probability that bead ib of the chain beingcomplexed with the macroion, will now be employed toobtain detailed information on the structure of thecomplex. The function possesses the lower boundaryPc(ib) ) 0, implying that bead ib is never complexed withthe macroion, and the upper one Pc(ib) ) 1, correspond-ing to the case that this bead is always complexed. Suchanalyses have previously provided valuable informationin studies30-32 on different systems containing polyionsand charged spherical colloids.

Figure 3 shows the complexation probabilities as afunction of ib for the cases displayed in Figure 1. Ideally,these probabilities should be symmetric with respect toa reflection at ib ) (Nb + 1)/2. This is fulfilled to a veryhigh degree, and the use of the biased slitheringtechnique to generate trial moves was in practicenecessary for such an achievement.

Starting with system I, Figure 3a displays how Pc(ib)evolves with increasing chain stiffness. For the fullyflexible chain (lP ) 7 Å), the complexation functiondisplays an extended plateau at ≈0.7 and a reducedprobability of complexation toward the ends of the chain.Such a plateau or several sections with high complex-ation probability will be referred to as binding region-(s). With increasing chain stiffness (lp ) 60 Å), theprobability of a bead in the binding region to becomplexed increases and the tails becomes more pro-nounced. With an even larger bare persistence length(lP ) 250 Å), three maxima appear signaling the ap-pearance of two loops, and with lP ) 500 and 1000 Åtwo well-separated maxima occur revealing one ex-tended loop. All these findings are consistent with theconfigurations shown in Figure 1.

Figure 3b provides the corresponding complexationfunctions for system II comprising a smaller macroion,Rm ) 10 Å, as compared to system I. Here, we observea similar behavior with increasing stiffness as in systemI, i.e., extended binding region for flexible chains andthe appearance of loops at increasing stiffness. However,three clear differences do also appear, viz. (i) in thecentral region Pc(ib) decreases continuously as lP isincreased, (ii) four maxima appear (lP ) 250 Å), indicat-ing the appearance of three loops, and (iii) threepronounced maxima occur (lP ) 500 Å), implying thepresence of two well-defined loops, all the differencesconsistent with the snapshots in Figure 1.

Regarding system III, the linear charge density ofthe polyion is twice as large and the length of thepolyion half as long as in system I. The correspondingcomplexation functions for system III are given inFigure 3c. With the fully flexible chain, we again ob-serve an extended binding region and short tails, herethe probability of a bead in the binding region to becomplexed being ≈0.95. As the chain becomes stiffer,(i) the complexation probability increases (as in sys-tem I), but remains near one even with the stiffestchain, and (ii) the tail becomes progressively longer.Obviously, the central part of the chain is alwaysnear the macroion surface, and no loops do appear.With the stiffest chain, the U-turn is composed of ca.20 beads.

Figure 2. Normalized running coordination number [rcn(r)/Nb] as a function of the distance from the center of themacroion (r) for systems (a) I, (b) II, (c) III, and (d) IV forindicated bare persistence lengths. Estimated errors aresmaller than the size of the symbols.


Finally, the corresponding data for system IV aregiven in Figure 3d. Throughout, most of the complexedbeads display a complexation probability near one. Withthe completely flexible chain, essentially all beads arecollapsed [Pc(ib) ≈ 1 for all ib], and with lP ) 60 Å onlythe end beads display a slight probability of being

detached. With an increased stiffness, the tails becomesprogressively longer. First with lP ) 1000 Å, do a singleloop appear, but even in this case considerable tails arepresent. The shortening of the binding region at in-creasing lp up to lp ) 500 Å is consistent with theprogressively unwrapping of the solenoid displayed inFigure 1.

Thus, the complexation probabilities provide a fin-gerprint of the chain configurations and confirm thatthe configurations shown in Figure 1 indeed representtypical structures of the systems. We also conclude thatin the present systems with equal absolute charge ofthe macromolecules the macroion prefer to complex tothe central part rather than to one end of the polyion(symmetric complexation).

3.2.3. Loop, Tail, and Train Characteristics. The chainconformations have furthermore been characterized byquantifying (i) the average number of beads in a loop,tail, and train, (ii) the average number of loops, tails,and trains, and (iii) the fraction of beads in loops, tails,and trains. Here, train refers to a contiguous sequenceof beads at the macroion surface, and in this analysiswe have used the same definition of Rc as above. Figure4 displays the number of beads in a feature of type R,⟨Nb⟩R, R ) (loop, tail, or train), as a function of the barepersistence length for the four systems, whereas theinsets of Figure 4 show the corresponding averagenumber of feature R, ⟨NR⟩ (except for ⟨Ntrain⟩, since ⟨Ntrain⟩≡ ⟨Nloop⟩ + 1). From ⟨Nb⟩R and ⟨NR⟩, the fraction of beadsin feature R is readily available through PR ) ⟨Nb⟩R⟨NR⟩/Nb.

Regarding systems I and II, Figure 4a quantifies theappearance of numerous short loops extending from thecollapsed layer with lP ) 7 Å, and the successivereduction of number of loops and the enlargement ofthe loops as the bare persistence length is increased.The fraction of beads being in a loop, Ploop, increasesmonotonically and displays a maximum for the stiffestchains (e.g., Ploop ) 0.62 for system II with lP ) 1000Å). Systems III and IV also display a reduction of thenumber of loops with increasing lP, but ⟨Nloop⟩ is smallalready for flexible chains. The length of the loopsappearing is short, just one or two beads. Hence, alsothe fraction of beads in loops is small. Regarding systemIV, the length of the single loop appearing at lP ) 1000Å is nearly 20 beads (half the chain).

Figure 4b shows that the number of tails and theaverage tail length, and hence also the fraction of beadsin tails, generally increase as the chain stiffness isincreased. Moreover, the four systems can also bedivided into the same two groups according to how thetail statistics depends on the bare persistence length.In systems I and II, ⟨Nb⟩tail and ⟨Ntail⟩ are larger thanthose in systems III and IV. With flexible chains,systems I and II possess chains with on the average tailscontaining ca. 3 beads, whereas in systems III and IVsuch tails lengths appears first at lP ≈ 250 Å. As thechain stiffness increases, ⟨Nb⟩tail in systems II and IIIincreases monotonically, displaying the maximum forthe stiffest chains. In systems I and IV, ⟨Nb⟩tail shows anonmonotonic behavior with the maximum at lP ≈ 500Å and decay for the stiffest chains considered. Finally,we notice that we generally do not have two tails. Withthe exception of system III, first with lP g 250 Å wefind ⟨Ntail⟩ > 1.95, demonstrating the nearly full devel-opment of two tails.

Figure 3. Complexation probability [Pc(ib)] as a function ofthe bead number (ib) for systems (a) I, (b) II, (c) III, and (d) IVwith bare persistence lengths lP ) 7 Å (filled squares), 60 Å(open squares), 250 Å (filled circles), 500 Å (open circles), and1000 Å (filled diamonds). Estimated errors are smaller thanthe size of the symbols, except for system II with lP g 60 Åwhere they are 0.05-0.09.


The number of beads in a train is given in Figure 4c.The general behavior is that ⟨Nb⟩train displays a maxi-mum at lP ≈ 100 Å; i.e., the longest contiguous sequenceof collapsed beads appears at intermediate stiffness.Also, here a 2-fold change in the macroion radius (cf.curves labeled I and II) has a smaller effect than a 2-foldchange in the linear charge density (cf. curves I and III).Systems I and II containing the longer chains but withlower linear charge density display a smaller numberof beads in a train as compared to systems III and IV.

A comparison between the data given in Figure 4 andthe snapshots in Figure 1 reveals again a consistencybetween the statistical analysis and the complexesshown.

3.3. Degree of Charge Neutralization. A centralquantity in polyion-macroion complexation is the ratioof charges carried by the polyion beads in the collapsedlayer and the macroion charge denoted by |Zp

c/Zm|. Thisquantity is often taken as a measure of the strength ofthe polyion-macroion complex. In our system with a1:1 complex and |Zp/Zm| ) 1, the upper limit of |Zp

c/Zm|

becomes unity, and the charge ratio is also equivalentto the fraction of beads in the collapsed layer.

Figure 5 shows the charge ratio |Zpc/Zm| as a function

of the bare persistence length for the four systems. Insystems I, III, and IV, the charge ratio displays aninitial rise, a maximum, and then a decrease as thestiffness is increased, whereas in system II, |Zp

c/Zm| is acontinuously decaying function. The maxima of thecharge ratio appearing in systems I, III, and IV are ofcourse related to the nonmonotonic rise of the reducedrcn with increasing lP (Figure 2) and the nonmonotonicvalue of Pc(ib) in the binding region with increasing lP

(Figure 3). The maxima in |Zpc/Zm| appear at lP ) 45,

65, and 30 Å (from harmonic fits), respectively. Insystems III and IV, |Zp

c/Zm| approaches nearly unity atits optimal lP expressing a nearly complete chargeneutralization, whereas in systems I and II the maximalcharge compensation approaches ca. 70 and 80%, re-spectively; thus, the smaller bead charge makes thecomplex weaker. System IV with the largest Γmb dis-plays a slightly larger maximum than system III, butthe degree of neutralization becomes the opposite atlarger chain stiffness. Obviously, with flexible chainsthe stronger macroion-bead attraction occurring withthe smaller macroion in system IV facilitates the chargeneutralization, whereas with stiffer chains, the largerelectrostatic interaction does not compensate the ener-getic bending penalty of having the chain to follow thesurface with the smaller curvature. In system III, evenat lP ) 1000 Å, ca. 50% of the beads remain in thecollapsed layer, consistent with the finding in section3.2.2 of having ca. 20 beads in the U-turn.

3.4. Energetics. Additional insight in the drivingforce of the complexation and structures attained isobtained by examining the contributions to the potentialenergy of the systems. Figure 6 displays the averageangular, electrostatic macroion-bead, and electrostaticbead-bead potential energy as a function of the barepersistence length for the four systems. In addition,corresponding electrostatic potential energies for twosimple models are given in Figure 6 (open symbols). Inthe first model (the collapsed model), all beads aredissociated and placed in hard-sphere contact with themacroion. Then, we have Umb/NbkT ) Γmb, whereas Ubb/NbkT was determined by numerical minimization of thetotal potential energy. In fact, Ubb/NbkT does not deviatemore than 20% from -Γmb/2 being the bead-beadenergy of having the bead charges homogeneously

Figure 4. Number of beads in (a) a loop (⟨Nb⟩loop), (b) a tail(⟨Nb⟩tail), and (c) a train (⟨Nb⟩train) as a function of the barepersistence length (lP) for indicated systems. The inserts in(a) and (b) display the number of loops ⟨Nloop⟩ and tails ⟨Ntail⟩,respectively. The lines are only guide for the eye.

Figure 5. Charge ratio (|Zpc/Zm|) as a function of the bare

persistence length (lP) for indicated systems. The lines are onlyguide for the eye. Estimated errors are smaller than the sizeof the symbols.


smeared out at r ) Rm + Rb. The second model involvesa rigid rod touching the macroion at the center of therod, and Umb/NbkT and Ubb/NbkT were calculated nu-merically by direct summation using rms bead-beadseparations from the simulations with the largest barepersistence length (see Table 2).

Figure 6a shows that the average angular energyinitially increases as the bare persistence length isincreased for all systems. We recall that the variationof the bare persistence length was achieved by alteringthe force constant of the angular potential energy termgiven by eq 5. However, at some (critical) stiffness theangular potential energy starts to decay, and thelocation of the maxima appear at 600, 330, 700, and 620Å (from harmonic fits) for systems I-IV, respectively.The angular energy at the critical stiffness is smallest

(≈1.8kT/Nb) for system I appearing when the polyionform a single, extended loop and largest (≈4kT/Nb) forsystem IV occurring when the polyion form a shortsolenoid (see Figure 1). We notice that the electrostaticmacroion-polyion attraction at contact is smallest forsystem I and largest for system IV.

Figure 6b shows that the average electrostatic mac-roion-bead interaction varies between ca. -3kT and-22kT. Below lP ≈ 100 Å, ⟨Umb⟩ is nearly constant, butat larger lP it starts to increase. A comparison between⟨Umb⟩ obtained from the flexible chain and the simplifiedcollapsed model shows that the former attains ca. 80-90% of the value of the latter. The smallest differencesappear for systems I and III with the larger macroion.Similarly, we found that ⟨Umb⟩ approaches its rod-limitas lP is increased.

The average electrostatic bead-bead repulsion isgiven in Figure 6c. Also, ⟨Ubb⟩ remains fairly constantup to lP ≈ 100 Å, although ⟨Ubb⟩ displays a gradualdecrease already at small lP in systems I and II. Thecomparison of ⟨Ubb⟩ from the flexible chain and thecollapsed model displays here a close agreement forsystems I and III and a considerable relative deviation(>25%) for systems II and IV. As lP increases, ⟨Ubb⟩approaches its rod limit.

Thus, energetically, the polyion sacrifices angularenergy to maintain the favorable electrostatic energyas the angular force constant is increased. However, atsome critical point, the bending energy becomes toolarge, and at this point the detachment of train beadsaccelerates. The bending energy at the critical pointincreases as the electrostatic macroion-bead attractionis increased.

4. Discussion

Our main findings will here be discussed across thedifferent systems and related to recent theoreticaldevelopments and MC simulations.

4.1. Flexible Polyion. System I with the lower linearcharged and longer polyion and the larger macroion (Zb) 0.5, Nb ) 80, and Rm ) 20 Å) possesses the weakestelectrostatic polyion-macroion interaction (Γmb ) 6.50).The polyion length is ca. 3 times the circumference ofthe macroion, and the bead coverage of the macroionsurface is 0.2 (see Table 2 for definitions). With the fullyflexible chain, the polyion forms a complex where thechain is locally folded, with short loops directed awayfrom the surface, making the charge ratio |Zp

c/Zm| dropto ca. 0.7. The low bead coverage makes the coverage ofthe macroion surface heterogeneous.

The specific parameters of system II are Zb ) 0.5, Nb) 80, and Rm ) 10 Å. As compared to system I, themacroion size is reduced by a factor of 2, giving anincreased electrostatic interaction (Γmb ) 11.9), anincreased polyion length to circumference length ratioto 6, and a bead coverage of 0.8. The bond coverage,which also is an appropriate measure of the surfacecoverage for chains with fixed or nearly fixed bondseparation, exceeds unity (Table 2). Figure 1 (row 1,column 2) clearly illustrates full surface coverage andthe inability for all beads to be in direct contact withthe macroion surface. A large amount of the beads formloops due to lack of space at the surface, and conse-quently the charge ratio is substantially reduced(|Zp

c/Zm| ) 0.78). The energy analysis also showed thatsystem II is the one deviating most from having a fullycollapsed bead layer.

Figure 6. Average and reduced (a) angular (⟨Uangle/NbkT⟩),(b) electrostatic macroion-bead (⟨Umb/NbkT⟩), and (c) electro-static bead-bead interactions (⟨Ubb/NbkT⟩) as a function of thebare persistence length (lP) for indicated systems (filledsymbols). The lines are only guide for the eye. In (b) and (c),the limiting interaction energies of a fully collapsed chain atthe macroion surface (open symbols, left) and of a rigid rodtouching the macroion (open symbols, right) are also given (seetext for further details). Estimated errors are smaller than thesize of the symbols.


System III is characterized by Zb ) 1, Nb ) 40, andRm ) 20 Å. The increase of the linear charge densityand reduction of the chain length with a factor of 2 ascompared to system I leads also to a stronger electro-static polyion-macroion interaction (Γmb ) 13.0). In thissystem, the flexible chain is again folded, but ascompared to system I, the stronger electrostatic interac-tion with the macroion makes the bead layer thinner,and the shorter chain is only able to make one turnaround the macroion. As in system I, large areas of themacroion are uncovered, and the charge ratio |Zp

c/Zm|increases to 0.93.

System IV possesses the strongest electrostatic in-teraction (Γmb ) 23.8) achieved by combing the highlinear charge density and the small macroion (Zb ) 1,Nb ) 40, and Rm ) 10 Å). The polyion length is againca. 3 times the circumference of the macroion and thebead and bond coverages amount to 0.4 and 0.7,respectively. Although possessing a high surface cover-age, the strong macroion-bead interaction is able tobring all beads near the surface and form a monolayerof beads without any protruding loops. The differentturns around the macroions are spatially correlated, butthe chain is still locally folded. The charge ratio |Zp

c/Zm|is nearly 1.0.

4.2. Nearly Flexible Chains. As expected, structuralrearrangements appear as the chain stiffness is in-creased. Already a moderate increase of the chainstiffness has consequences on the complex. The sup-pression of the local folding by a weak angular potentialincreasing the bare persistence length from 7 to 30-60Å enables the beads to approach closer to the macroionas deduced form, e.g., the reduced rcn and the chargedratio |Zp

c/Zm|. Obviously, for a fully flexible chain, theelectrostatically driven collapse is partly counteractedby chain configuration entropy. System II constituteshowever an exception, since the macroion area is notsufficiently large to house all beads. In system III, thechain attains an incomplete “tennis ball seam” confor-mation (the chain is too short to make a full seam),whereas in system IV with the smaller macroion athree-turn solenoid is formed. In systems III and IV,essentially no tails appear yet.

4.3. Stiffer Chains. When the stiffness of the chainsis increased further, additional structural rearrange-ments appear. The difference in the macroion sizebecomes less important as compared to the differencein chain linear charge density (and length). As the barepersistence length exceeds the macroion curvature, theangular potential starts to markedly affect the abilityof the chain to be collapsed at the macroion.

In systems I and II, more extended loops are formed,and in particular in system II with the smaller macroionand stronger bead-macroion interaction “rosette”-likestructures are formed. At even larger bare persistencelengths, the angular energy exceeds the gain in elec-trostatic energy of forming many chain-macroion con-tacts. At this stage the chain releases bending tensionby reducing the number of contact points and becomesstraighter. Finally, with an even larger bare persistencelength (or weaker electrostatic interaction) than con-sidered here, the polyions would straighten out evenfurther and only a single polyion-macroion contactregion would appear as the equilibrium configuration.

The structural developments in systems III and IVare however different. In system III, the chain isstraightened further, albeit still being collapsed at the

macroion surface by forming a one and half turn aroundthe macroion. First with lP ) 500 Å, substantial tailsare formed and with the stiffest chain considered, aU-turn involving still half of the chain is formed. Insystem IV, the ends of the three-turn solenoid start todetach as the stiffness is increased. Between lP ) 500and 1000 Å there is a transition to a single-loopconformation. At this stage the bending tension becomestoo large to keep the chain folded around the macroion.With an even larger stiffness, also here a single poly-ion-macroion contact region would appear.

4.4. Relation to Theories and Previous ModelSimulations. Despite the large number of theoriesdeveloped to describe the polyion-macroion complex-ation, yet not a single theory covers the rich complexbehavior explored here by MC simulations.

The collapsed state of flexible polyions has beenexamined by a number of theories,20-22,26,33 and inparticular the aspect of overcharging has been consid-ered. However, bead-bead excluded-volume effects aregenerally ignored, and the present study involves asystem with a high surface coverage (system II), wheresuch an effect plays a significant role. We have alsodemonstrated the effect of the chain configurationalentropy and showed that the strongest complexation isobtained with a semiflexible chain. This is not toosurprising, since the adsorption of polyions to an op-positely charged plane improves with increasing chainstiffness.

The spatial arrangements of semiflexible chainsinteracting with a small spherical center through ashort-range attractive force have theoretically beentreated by Schiessel et al.25 They employed a wormlikechain model, which describes the chain as a semiflexibletube characterized by a bending and a torsional stiff-ness. The theoretically treatment required them toconsider a ring polymer, but the results are not claimedto depend critically on an opening of the ring.25 At lowbending stiffness and at not too attractive chain-sphereinteraction, their theory predicts a “rosette”-like struc-ture (see Figure 1 of ref 25) with up to five loops andwith the number of loops reducing as the bendingstiffness was increased. Both the appearance of loopsand the reduction of the number of them are inqualitatively agreement with our predictions for systemII. Moreover, the theory predicts a first-order transitionto a compact wrapping state (see Figure 2 of ref 25) asthe attraction between the tube and the center isincreased for a given bending stiffness. Similar transi-tion is found here as the linear charge density of thepolyion is increased (see Figure 1, row 3, column 2 fcolumn 4), but the location of this transition has nothere been investigated in further detail.

Recently, Netz and Joanny modeled the polyion-macroion complexation using a continuous polymermodel containing bending energy, repulsive electrostaticintrachain interaction, and attractive electrostatic in-teraction between the polymer and the macroion.23 Theground state approximation was invoked making thetheory only applicable for bare persistence lengthslarger than the size of the macroion. Moreover, thepolymer was restricted to a common plane with themacroion center, and the electrostatic interaction wastreated on the Debye-Huckel level. Of interest here,they derived scaling relations for the “touching” and“wrapping” transitions. The former transition was de-fined as the point where the chain locally attain the


curvature of the macroion and the latter where thelength of the wrapping polyion segment approaches theradius of the macroion. Owing to (i) different polymermodels and (ii) screened vs no screened electrostaticinteraction detailed comparison is impeded. Neverthe-less, for a constant screening and flexibility Netz andJoanny predicted transitions from point contact, looselywrapped, and to tightly wrapped complexes as themacroion charge is increased (Figure 5 of ref 23), similarto our observations and conclusions.

Finally, Chodanowski and Stoll have performed ex-tensive MC simulation of the complexation between onepolyion and an oppositely charged macroion interactingvia the screened Debye-Huckel potential using a freelyjointed hard-sphere chain.28,29 Their studies involvedsimulation of one chain with varying chain lengths(25-200 beads with fixed linear charged density-1e/7.14 Å) and one macroion (Rm ) 17.85 Å and Zm )50)28 and one chain (100 beads) and one macroion withvariable size and charge at fixed surface charge den-sity,29 both cases at different screening lengths (0-1 M1:1 salt) and were focused on the adsorption/desorptiontransition and the structure of the complexes. Themacroion size and linear charge density used28 areembraced by the values used here, and at those fewconditions where our model parameters are similar(their Nb ≈ 50), the obtained wrapping structure issimilar. From their simulations with variable screeninglength, they concluded that the adsorption/desorptiontransition appears at a macroion-bead interaction of≈1kT, supporting the present observation of strongcomplexation.

5. ConclusionOn the basis of Monte Carlo simulations, an exami-

nation of the complexation between a single polyion anda single, oppositely charged, and spherical macroion hasbeen performed. Relevant physical parameters havebeen discussed, and focus has been made on how thebare persistence length affects the structure and ener-getics. This was made for some combinations of polyionlinear charge densities and macroion sizes at constantand equal absolute polyion and macroion charge withoutsalt applied to aqueous solution at ambient tempera-ture. Despite the selected strong polyion-macroioninteraction (O(10kT/bead)) and the concomitant largeenergy barriers between different states, equilibriumresults were obtained by combining different types oftrial moves and in some cases considering differentinitial configurations.

The flexible chains displayed a collapsed state withthe chains wrapped around the macroion but still locallyfolded. The maximal collapse appeared for a semiflexiblechain. The fully flexible chains displayed a locally morefolded configuration opposing an optimal collapse. Alarge range of structures was encountered as the chainstiffness gradually was increased. The collapsed stateappearing with a flexible chain was successively trans-formed into “tennis ball seam”, solenoid, multiloop(“rosette”-like), single-loop, and U-turn structures, how-ever not all of them present in the same system. Thenumber of beads in a single loop and tail was increasedas the chain stiffness was increased, whereas thenumber of beads in a single train displayed a maximumat intermediate stiffness. In the present systems withequal absolute charge of the macromolecules the mac-roion prefer to complex to the central part rather thanto one end of the polyion (symmetric complexation).

The structure of the present system is primarilycontrolled by (i) the attractive polyion-macroion inter-action favoring the complexation and (ii) the repulsivebead-bead interaction and the bending interactionopposing the complexation. The present study givesadditional insight into the magnitudes of these interac-tion terms as well as the competition among them.

Although quantitative comparison with recent theo-retical developments is no yet trivial due to differentmodel systems and features studied, important struc-tures previously predicted have here been confirmed.

Acknowledgment. A.A. thanks Andrei Broukhnoand Magnus Ullner for helpful discussions. Financialsupport from the Swedish National Research Council(NFR) is gratefully acknowledged.

AppendixA short account of different approaches to evaluate

the persistence length of a chain composed of a sequenceof beads and a motivation of our choice are given here.

The persistence length lP is used as a measure of thechain stiffness, and it has been defined and evaluatedin several different ways.39,40 Using global chain proper-ties, the persistence length can be defined by

where Ree is the end-to-end distance, Nb the number ofbeads in the chain, and b the bond length. In anotherglobal approach based on the wormlike chain model, lPis given by

with the limiting value

as L/lP,RG f ∞, where in eqs 8 and 9 RG denotes theradius of gyration and L ) (Nb - 1)b the polymercontour length. Since L/lp,RG is typically large in experi-mental systems and RG is readily available from e.g.scattering experiments, eq 9 is often used when report-

Figure 7. Bare persistence length (lP) as a function of thenumber of beads (Nb) of an uncharged chain of Nb beads joinedby a harmonic bond potential given by eq 4 with kbond ) 0.4N/m and r0 ) 5.0 Å and by a harmonic angle potential givenby eq 5 with kangle ) 35 J/(mol deg2) and R0 ) 180°. The barepersistence length was evaluated using indicated equations.

lP,Ree)

Ree2

2(Nb - 1)b+ b

2(7)

RG2 )

LlP,RG

3- lP,RG

2 + 2lP,RG

3

L- 2

lP,RG

4

L2(1 - e-L/lP,RG) (8)

lP,RGf 3RG

2 /L (9)


ing flexibility of experimental polymer systems. Equa-tions 7 and 9 are equivalent for a Gaussian chain where⟨RG

2 ⟩ ) ⟨Ree2 ⟩/6.

In a third approach, the persistence length is basedon the projection of angles between bonds vectorsaccording to

where b is again the bond length and θk is the anglebetween two bond vectors separated by k bonds. For afreely rotating chain with equal and fixed angles, theaverage of directional cosines becomes ⟨cos θk⟩ )cosk θ1, where θ1 is the first directional angle. The sumin eq 10 becomes a geometric series, and for an infinitechain we obtain

In the case of fluctuating bond length and bond angle,we substitute b by ⟨Rbb

2 ⟩1/2 and cos θ1 by ⟨cos θ1⟩.Moreover, the substitution θ1 ) π - R leads to thefollowing expression

Figure 7 shows the persistence length of an unchargedchain of beads connected by harmonic bonds as obtainedusing eqs 7, 8, and 12 as a function of the number ofbeads using kangle ) 35 J/(mol deg2). The persistencelength of all three approaches appears to converge to lP≈ 260 Å as Nb is increased; however, the rate ofconvergence differs. The route using the end-to-endseparation has the slowest convergence (still not con-verged at Nb ) 500 or L ≈ 10lP) and approaches its limitfrom below. The use of the radius of gyration convergesmuch faster. Finally, the local approach is, as expected,least sensitive on the chain length. Since, in this studyvery rigid chains with lP . L are considered, the localapproach of calculating the persistence length has beenadopted. Here, we do not concern ourselves about theinterpretations of the different methods of evaluatingpersistence lengths.39,40

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MA012052U

lP,proj ) b∑k

⟨cos θk⟩ (10)

lP,proj ) b1 - cos θ1

(11)

lP,proj )⟨Rbb

2 ⟩1/2

1 + ⟨cos R⟩(12)


monte carlo simulations of polyion−macroion complexes. 1. equal absolute polyion and macroion...

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